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Datei: angles_2D_scaf.prf   Sprache: Lisp

Original von: PVS©

(angles_2D_scaf
 (trig_prop2 0
  (trig_prop2-1 nil 3408794701
   ("" (skeep)
    (("" (cross-mult -1)
      (("" (lemma "sq_eq_abs")
        (("" (inst - "sin(a) * u`x" "u`y * cos(a)")
          (("" (flatten)
            (("" (hide -1)
              (("" (split -1)
                (("1" (rewrite "sq_times")
                  (("1" (rewrite "sq_times")
                    (("1" (copy -1)
                      (("1" (lemma "sin2")
                        (("1" (inst?)
                          (("1" (replace -1 -2)
                            (("1" (assert)
                              (("1"
                                (move-terms -2 l 2)
                                (("1"
                                  (hide -1)
                                  (("1"
                                    (lemma "cos2")
                                    (("1"
                                      (inst?)
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                                                    (("1"
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                                                        (assert)
                                                        nil
                                                        nil))
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                                                      (hide 4)
                                                      (("2"
                                                        (lemma
                                                         "dot_sq_norm")
                                                        (("2"
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                                                          (("2"
                                                            (expand
                                                             "*")
                                                            (("2"
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                                                               "sq")
                                                              (("2"
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                                                                nil
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    nil)
   ((real_div_nzreal_is_real application-judgement "real" reals nil)
    (div_cancel3 formula-decl nil real_props nil)
    (Nz_vect2 type-eq-decl nil vectors_2D nil)
    (zero const-decl "Vector" vectors_2D nil)
    (Vector type-eq-decl nil vectors_2D nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (sin const-decl "real" trig_basic "trig/")
    (cos const-decl "real" trig_basic "trig/")
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (/= const-decl "boolean" notequal nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (cos_range application-judgement "trig_range" trig_basic "trig/")
    (real_times_real_is_real application-judgement "real" reals nil)
    (times_div2 formula-decl nil real_props nil)
    (sin_range application-judgement "trig_range" trig_basic "trig/")
    (div_cancel4 formula-decl nil real_props nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sq_times formula-decl nil sq "reals/")
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (nz_norm_gt_0 application-judgement "posreal" vectors_2D nil)
    (real_minus_real_is_real application-judgement "real" reals nil)
    (* const-decl "real" vectors_2D nil)
    (dot_sq_norm formula-decl nil vectors_2D nil)
    (nnreal type-eq-decl nil real_types nil)
    (norm const-decl "nnreal" vectors_2D nil)
    (cos2 formula-decl nil trig_basic "trig/")
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (IFF const-decl "[bool, bool -> bool]" booleans nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (- const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (>= const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sin2 formula-decl nil trig_basic "trig/")
    (sq_eq_abs formula-decl nil sq "reals/"))
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                       ("2" (lemma "sq_eq")
                        (("2" (inst - "-u`x" " norm(u) * cos(a)")
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                                    nil
                                    nil))
                                  nil))
                                nil))
                              nil))
                            nil)
                           ("2" (lemma "sq_eq")
                            (("2" (inst - "-u`x" "-norm(u) * cos(a)")
                              (("1"
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                                      (("1"
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                                              nil
                                              nil))
                                            nil))
                                          nil))
                                        nil))
                                      nil))
                                    nil))
                                  nil))
                                nil)
                               ("2" (assertnil nil)
                               ("3" (assertnil nil))
                              nil))
                            nil)
                           ("3" (assertnil nil))
                          nil))
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                    nil))
                  nil))
                nil))
              nil))
            nil))
          nil))
        nil))
      nil))
    nil)
   ((trig_prop2 formula-decl nil angles_2D_scaf nil)
    (minus_real_is_real application-judgement "real" reals nil)
    (nz_norm_gt_0 application-judgement "posreal" vectors_2D nil)
    (real_div_nzreal_is_real application-judgement "real" reals nil)
    (cos const-decl "real" trig_basic "trig/")
    (norm const-decl "nnreal" vectors_2D nil)
    (nnreal type-eq-decl nil real_types nil)
    (>= const-decl "bool" reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (sq_times formula-decl nil sq "reals/")
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (u skolem-const-decl "Nz_vect2" angles_2D_scaf nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (a skolem-const-decl "real" angles_2D_scaf nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (= const-decl "[T, T -> boolean]" equalities nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (<= const-decl "bool" reals nil)
    (- const-decl "[numfield -> numfield]" number_fields nil)
    (trig_range type-eq-decl nil trig_basic "trig/")
    (nzreal_div_nzreal_is_nzreal application-judgement "nzreal"
     real_types nil)
    (both_sides_times1_imp formula-decl nil extra_real_props nil)
    (nznum nonempty-type-eq-decl nil number_fields nil)
    (/ const-decl "[numfield, nznum -> numfield]" number_fields nil)
    (CBD_14 skolem-const-decl "trig_range" angles_2D_scaf nil)
    (bijective? const-decl "bool" functions nil)
    (id const-decl "(bijective?[T, T])" identity nil)
    (TRUE const-decl "bool" booleans nil)
    (minus_nzreal_is_nzreal application-judgement "nzreal" real_types
     nil)
    (sq_neg formula-decl nil sq "reals/")
    (minus_odd_is_odd application-judgement "odd_int" integers nil)
    (neg_one_times formula-decl nil extra_tegies nil)
    (CBD_15 skolem-const-decl "trig_range" angles_2D_scaf nil)
    (sq_eq formula-decl nil sq "reals/")
    (CBD_16 skolem-const-decl "trig_range" angles_2D_scaf nil)
    (neg_neg formula-decl nil extra_tegies nil)
    (mult_neg formula-decl nil extra_tegies nil)
    (neg_mult formula-decl nil extra_tegies nil)
    (CBD_17 skolem-const-decl "trig_range" angles_2D_scaf nil)
    (sin const-decl "real" trig_basic "trig/")
    (real_times_real_is_real application-judgement "real" reals nil)
    (cross_mult formula-decl nil real_props nil)
    (Nz_vect2 type-eq-decl nil vectors_2D nil)
    (zero const-decl "Vector" vectors_2D nil)
    (/= const-decl "boolean" notequal nil)
    (Vector type-eq-decl nil vectors_2D nil)
    (real nonempty-type-from-decl nil reals nil)
    (real_pred const-decl "[number_field -> boolean]" reals nil)
    (number_field nonempty-type-from-decl nil number_fields nil)
    (number_field_pred const-decl "[number -> boolean]" number_fields
     nil)
    (boolean nonempty-type-decl nil booleans nil)
    (number nonempty-type-decl nil numbers nil)
    (sin_range application-judgement "trig_range" trig_basic "trig/")
    (cos_range application-judgement "trig_range" trig_basic "trig/")
    (sq_nz_pos application-judgement "posreal" sq "reals/"))
   nil)
  (trig_prop-1 nil 3408734330
   ("" (skeep)
    (("" (cross-mult -1)
      (("" (lemma "sq_eq")
        (("" (inst?)
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                      (("1" (copy -1)
                        (("1" (lemma "sin2")
                          (("1" (inst?)
                            (("1" (replace -1 -2)
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                                (assert)
                                (("1"
                                  (move-terms -2 l 2)
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                                    (("1"
                                      (lemma "cos2")
                                      (("1"
                                        (inst?)
                                        (("1"
                                          (replace -1 -3)
                                          (("1"
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                                                (move-terms -2 r 2)
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                                                  (factor -1)
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                                                    (factor -2)
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                                                       "sq(v`y) + sq(v`x) = sq(norm(v))")
                                                      (("1"
                                                        (hide -1)
                                                        (("1"
                                                          (rewrite
                                                           "sq_times"
                                                           :dir
                                                           rl)
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                                                             "sq_times"
                                                             :dir
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                                                                 -1)
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                                                               ("2"
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                                                                 -2)
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                                                       ("2"
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                                                                          (("1"
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      (("1" (case-replace "pa * sin(alpha) = pb * sin(beta)")
        (("1" (case-replace "pa = pb ")
          (("1" (lemma "sin_cos_eq")
            (("1" (inst - "alpha" "beta")
              (("1" (assert)
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           ("2" (hide 2)
            (("2"
              (case " sq(pa) * sq(cos(alpha)) +
                                        sq(pa) * sq(sin(alpha)) =  sq(pb) * sq(cos(beta))
                                                                  + sq(pb) * sq(sin(beta))")
              (("1" (hide -2 -3)
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               ("2" (rewrite "sq_times" :dir rl)
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                  (("2" (rewrite "sq_times" :dir rl)
                    (("2" (rewrite "sq_times" :dir rl)
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     nil)
    (numfield nonempty-type-eq-decl nil number_fields nil)
    (* const-decl "[numfield, numfield -> numfield]" number_fields nil)
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    (real nonempty-type-from-decl nil reals nil)
    (bool nonempty-type-eq-decl nil booleans nil)
    (>= const-decl "bool" reals nil)
    (nonneg_real nonempty-type-eq-decl nil real_types nil)
    (> const-decl "bool" reals nil)
    (posreal nonempty-type-eq-decl nil real_types nil)
    (cos const-decl "real" trig_basic "trig/")
    (nnreal type-eq-decl nil real_types nil)
    (< const-decl "bool" reals nil)
    (AND const-decl "[bool, bool -> bool]" booleans nil)
    (pi_lb const-decl "posreal" trig_basic "trig/")
    (pi_ub const-decl "posreal" trig_basic "trig/")
    (pi const-decl "{r: posreal | r > pi_lb AND r < pi_ub}" trig_basic
     "trig/")
    (nnreal_lt_2pi nonempty-type-eq-decl nil atan2 "trig/")
    (sin_range application-judgement "trig_range" trig_basic "trig/")
    (cos_range application-judgement "trig_range" trig_basic "trig/")
    (nonzero_real nonempty-type-eq-decl nil reals nil)
    (/= const-decl "boolean" notequal nil)
    (both_sides_times1 formula-decl nil real_props nil)
    (sin_cos_eq formula-decl nil trig_extra "trig/")
    (nnreal_plus_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (+ const-decl "[numfield, numfield -> numfield]" number_fields nil)
    (sq const-decl "nonneg_real" sq "reals/")
    (nnreal_times_nnreal_is_nnreal application-judgement "nnreal"
     real_types nil)
    (sq_nz_pos application-judgement "posreal" sq "reals/")
    (abs const-decl "{n: nonneg_real | n >= m AND n >= -m}" real_defs
         nil)
    (posreal_times_posreal_is_posreal application-judgement "posreal"
     real_types nil)
    (nonzero_abs_is_pos application-judgement "{y: posreal | y >= x}"
     real_defs nil)
    (real_ge_is_total_order name-judgement "(total_order?[real])"
     real_props nil)
    (sq_eq_abs formula-decl nil sq "reals/")
    (sin2_cos2 formula-decl nil trig_basic "trig/")
    (sq_times formula-decl nil sq "reals/")
    (sin const-decl "real" trig_basic "trig/"))
   shostak)))


¤ Dauer der Verarbeitung: 0.34 Sekunden  (vorverarbeitet)  ¤





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